nLab Riemann hypothesis

Idea

The Riemann hypothesis or Riemann conjecture is the famous unproved statement that all nontrivial zeros of the Riemann zeta function are on the vertical line $Re(z)=1/2$ in the complex plane.

Analogues of the Riemann hypothesis can be considered for many analogues of zeta functions and L-functions, here one speaks of generalized Riemann hypotheses.

An important special case over finite fields is called the Riemann–Weil conjecture and was proved by Deligne building on earlier ideas of Weil and Grothendieck. Grothendieck however expected a more natural proof using the (hypothetical) theory of motives.

Related concepts

• Weil conjecture

• Selberg trace formula

• Riemann hypothesis and physics

References

General

• Terence Tao, The Riemann hypothesis in various settings

See also

• Wikipedia, Riemann hypothesis

Proof strategies

The suggestion that the Riemann hypothesis might have a proof that is an analogue of Weil’s proof for arithmetic curves over finite fields $\mathbb{F}_q$ but generalized to the field with one element is due to

• Yuri Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa) Asterisque, (228):4, 121-163, 1995. Columbia University Number Theory Seminar.

• Wikipedia, Generalized Riemann hypothesis.

Identification of the zeros of the Riemann zeta function with the spectrum of p-adic string theory is due to

• An Huang, Bogdan Stoica, Shing-Tung Yau, General relativity from $p$-adic strings (arXiv:1901.02013)

Computer-checks

Experimental checks of the Riemann hypothesis with computers:

• R. P. Brent, J. van de Lune, H. J. J. te Riele and D. T. Winter, On the Zeros of the Riemann Zeta Function in the Critical Strip. II, Mathematics of Computation Vol. 39, No. 160 (Oct., 1982), pp. 681-688 (doi:10.2307/2007345)

• A. M. Odlyzko, The $10^{22}$-nd zero of the Riemann zeta function, in Lapidus et al. (eds.) proceedings of AMS special session on Dynamical, Spectral, and Arithmetic Zeta Functions, 1999 (pdf)

• Xavier Gourdon, The $10^{13}$ first zeros of the Riemann Zeta function, and zeros computation at very large height, 2004 (pdf, webpage)

• David J. Platt, Isolating some non-trivial zeros of zeta, Mathematics of Computation 86 (2017), 2449-2467 (doi:10.1090/mcom/3198)